For the Hamiltonicity of balanced bipartite graphs,
Li and Ning [1]} also proved the spectral version of
the Moon–Moser Theorem REF , that is,
the bipartite version of Theorems REF  and REF .
Later,
Ge and Ning [2]},
and Jiang, Yu and Fang [3]} independently
proved a further improvement on the
adjacency spectral result of Li and Ning for
balanced bipartite graphs.
Correspondingly,
Li, Liu and Peng [4]}
also gave a further improvement
on the signless Laplacian spectral result of Li and Ning.
It is noteworthy that Liu, Wu and Lai [5]}
unified several former spectral Hamiltonian results
on balanced bipartite graphs.
In addition, Lu [6]}
extended some spectral conditions for the Hamiltonicity
of balanced bipartite graphs.
It is possible ponderable to extend our results in
Section  into the balanced bipartite graphs. 
